Introduction to advanced quantum chemistry fileIntroduction to advanced quantum chemistry Introduction to advanced quantum chemistry Emmanuel Fromager Institut de Chimie de Strasbourg

Introduction to advanced quantum chemistry


Emmanuel Fromager

Institut de Chimie de Strasbourg - Laboratoire de Chimie Quantique -Université de Strasbourg /CNRS

M2 lecture, Strasbourg, France.

Institut de Chimie, Strasbourg, France Page 1


Electronic Hamiltonian, SI and atomic units

• We work within the Born–Oppenheimer approximation (the nuclei are fixed)

• The N -electron Hamiltonian can be written as

H = T + Wee + Vne

T ≡N∑i=1

−~2

2me∇2

ri=

N∑i=1

−~2

2me

(∂2

∂x2i

+∂2

∂y2i

+∂2

∂z2i

)→ kinetic energy

Wee =1

2

N∑i 6=j

wee(i, j) with wee(i, j) ≡e2

4πε0rij× → electron-electron repulsion

Vne =

N∑i=1

vne(i) with vne(i) ≡ −nuclei∑

A

ZAe2

4πε0|ri −RA|× → electron-nuclei attraction

• A physical N -electron wavefunction Ψ(r1, r2, . . . , rN ) depends on the positions of each electron

(spin will be introduced later on) and fulfils the Schrödinger equation HΨ = EΨ .




• Hydrogen atom (N = 1):

H → −~2

2me

(∂2

∂x2+

∂2

∂y2+

∂2

∂z2

)−

e2

4πε0√x2 + y2 + z2

× ,

E → En = −EI

n2where the ionization energy equals EI =

mee4

2(4πε0)2~2≈ 13.6 eV.

The ground-state wavefunction (n = 1) equals Ψ1s(x, y, z) =1

√πa

3/20

e−√

x2+y2+z2/a0 where

the Bohr radius equals a0 =4πε0~2

mee2≈ 0.529 Å.

• Working with so-called "atomic units" simply consists in using unitless energy E = E/2EI andcoordinates x = x/a0, y = y/a0, z = z/a0.

• The ground-state energy of the hydrogen atom is therefore −0.5 in atomic units.

• Returning to the general N -electron problem, the Schrödinger equation in atomic units is obtained

fromHΨ

2EI= EΨ




• Change of variables in the wavefunction:

Ψ(r1, r2, . . . , rN ) = Ψ(a0r1, a0r2, . . . , a0rN ) = Ψ(r1, r2, . . . , rN ) = Ψ

(r1

a0,r2

a0, . . . ,

rN

a0

)

Using Ψ rather than Ψ and the relations 2EI =~2

mea20

=e2

4πε0a0leads to

T /2EI ≡N∑i=1

−a2

0

2

(∂2

∂x2i

+∂2

∂y2i

+∂2

∂z2i

)≡

N∑i=1

−1

2

(∂2

∂x2i

+∂2

∂y2i

+∂2

∂z2i

),

Wee/2EI ≡1

2

N∑i 6=j

a0

rij× =

1

2

N∑i 6=j

1

rij×,

Vne/2EI ≡N∑i=1

−ZAa0

|ri −RA|× =

N∑i=1

−ZA

|ri − RA|×

• In the following we will simply drop the "tilde" symbol and denote T /2EI as T , Wee/2EI as Wee,Vne/2EI as Vne, and H/2EI as H .



Connecting theory and experiment

• The quantities we will mainly focus on are the ground-state E0 and excited-state {EI}I>0 energies.

• We will assume for simplicity that the ground state is not degenerate like, for example, in the heliumatom:

E0 1s2

E1 = E2 = E3 1s2s (triplet)E4 1s2s (singlet)

• Electronic excitation energies ωI = EI − E0 can be measured (UV/visible spectroscopy)

• The ground-state energy E0 alone is already very interesting. Indeed, it gives access to equilibriumgeometries, vibrational frequencies, static response properties (polarizabilities, magneticsusceptibilities, ...)



Vibrational frequencies in diatomics

• Let us return to the general problem where both nuclei and electrons are treatedquantum-mechanically. For simplicity, we will ignore rotation and assume that the mass M ofnucleus B is much larger than the mass m of nucleus A (m << M ) thus leading to the Schrödingerequation for the molecule

HmolΨmol(R,q) = EmolΨmol(R,q)

where the molecular Hamiltonian equals in atomic units

Hmol ≡ −1

2m

d2

dR2+ZAZB

R+ H(R).

R is the distance between A and B, H(R) is the electronic Hamiltonian (simply referred to as H onprevious slides) that depends explicitly on R, and q ≡ (r1, r2, . . . , rN ) is the position vector forall the electrons.

• Within the Born-Oppenheimer approximation, the molecular wavefunction is decomposed asfollows, Ψmol(R,q) = χ(R)Ψ(R,q), where

H(R)Ψ(R,q) = E(R)Ψ(R,q)



• Within the so-called adiabatic approximation, variations of the electronic wavefunction with nucleardisplacements are neglected: ∂Ψ(R,q)/∂R ≈ 0 ≈ ∂2Ψ(R,q)/∂R2.

• Such an approximation is in principle relevant when the molecule is at equilibrium (R does not varysignificantly).

• Therefore, the nuclear wavefunction fulfills the following Schrödinger equation[−

1

2m

d2

dR2+ V (R)×

]χ(R) = Emolχ(R),

where V (R) = E(R) +ZAZB

R.

• V (R) is the potential interaction energy between the two nuclei in the field of the electrons.

• From a classical mechanics point of view, the force F (R) = −dV (R)/dR is applied to nucleus A.

• The equilibrium distance R0 is such that F (R) < 0 when R > R0 and F (R) > 0 when R < R0.

• F (R) being continuous implies F (R0) = 0 =dV (R)

dR

∣∣∣∣R=R0

= 0 ← equilibrium structure !



• Note that an equilibrium structure is indeed found when R0 corresponds to a minimum for V (R).

• In the general case (larger molecules), R will be a reaction coordinate and maxima of V (R) willcorrespond to transition states. Once reached, the molecule will "move" to another stable state thatcorresponds to a local minimum for V (R).

• Let us consider fluctuations x = R−R0 around the equilibrium bond distance R0. From the Taylorexpansion through second order in x,

V (R) = V (R0 + x) ≈ V (R0) +1

2

(d2V (R)

dR2

∣∣∣∣R=R0

)x2,

we recover the Schrödinger equation for the harmonic oscillator with frequency

ω =

√1

m

d2V (R)

dR2

∣∣∣∣R=R0

or, equivalently, with constant k =d2V (R)

dR2

∣∣∣∣R=R0

,

[−

1

2m

d2

dx2+

1

2mω2x2×

]ϕ(x) =

(Emol − V (R0)

)ϕ(x)

where ϕ(x) = χ(R0 + x).

• This approximation is known as the harmonic approximation.



• The exact solutions to this problem are known: Emoln = V (R0) + ω

(n+

1

2

)where n = 0, 1, 2, . . .

• The vibrational frequency ω can be measured by infrared spectroscopy.

• The equilibrium bond distance R0 can be measured, for example, by microwave spectroscopy(rotational spectroscopy).

• When the molecule is in its ground vibrational state (n = 0) its energy equals

Emol0 = V (R0) +

ω

2︸︷︷︸zero point energy (ZPE)

• The binding energy can then be decomposed as follows

D0 = V (+∞)− V (R0)︸︷︷︸−ω2De



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Hellmann–Feynman theorem• Let us consider the electronic Schrödinger equation

H(Q)|Ψ(Q)〉 = E(Q)|Ψ(Q)〉

where Q = (Q1, Q2, . . .) is a collection of parameters (which, of course, does not include q!).

• The Q-dependent eigenvector |Ψ(Q)〉 is normalized for any Q: 〈Ψ(Q)|Ψ(Q)〉 = 1 .

• The Hellmann–Feynman theorem states thatdE(Q)

dQi=

⟨Ψ(Q)

∣∣∣∣∣∂H(Q)

∂Qi

∣∣∣∣∣Ψ(Q)

⟩.

Proof:

dE(Q)

dQi=

⟨Ψ(Q)

∣∣∣∣∣∂H(Q)

∂Qi

∣∣∣∣∣Ψ(Q)

⟩+

⟨∂Ψ(Q)

∂Qi

∣∣∣∣H(Q)

∣∣∣∣Ψ(Q)

⟩+

⟨Ψ(Q)

∣∣∣∣H(Q)

∣∣∣∣∂Ψ(Q)

∂Qi

⟩︸︷︷︸

E(Q)d

dQi

[〈Ψ(Q)|Ψ(Q)〉

]= 0



• Application to the calculation of molecular forces: Q = {RB}B ← position vectors of the nuclei !

The electronic Hamiltonian equals (in atomic units)

H(Q) ≡N∑i=1

−1

2∇2

ri+

1

2

N∑i6=j

1

|ri − rj |× −

N∑i=1

nuclei∑B

ZB

|ri −RB |×

thus leading to

∂H(Q)

∂RA≡ −

N∑i=1

ZA(ri −RA)

|ri −RA|3× .

The total energy (including nuclear-nuclear repulsions) reads

V (Q) = E(Q) +

nuclei∑B<C

ZBZC

|RB −RC |

so that, according to the Hellmann–Feynman theorem, the force applied to nucleus A equals

FA(Q) = −dV (Q)

dRA=

⟨Ψ(Q)

∣∣∣∣∣N∑i=1

ZA(ri −RA)

|ri −RA|3×

∣∣∣∣∣Ψ(Q)

⟩+

nuclei∑B 6=A

ZAZB(RA −RB)

|RA −RB |3



• Note that

one-electron operator

⟨Ψ(Q)

∣∣∣∣∣∣∣∣︷︸︸︷N∑i=1

ZA(ri −RA)

|ri −RA|3×

∣∣∣∣∣∣∣∣Ψ(Q)

⟩

=

∫dr1 . . .

∫drN Ψ∗(Q, r1, . . . , rN )

N∑i=1

ZA(ri −RA)

|ri −RA|3×︷︸︸︷Ψ(Q, r1, . . . , rN )

=

N∑i=1

∫dri (−1)×

[∏j 6=i

∫drj |Ψ(Q, r1, . . . , rN )|2

]︸︷︷︸

ZARA − ri

|ri −RA|3

density of charge for electron i at position ri

electronic wavefunction



Variational principle for the ground state

• Let {ΨI}I=0,1,2,... denote the exact orthonormal electronic ground-state (I = 0) and excited-state(I > 0) wavefunctions:

H|ΨI〉 = EI |ΨI〉, 〈ΨI |ΨJ 〉 = δIJ

• We assume for clarity that the ground state is non-degenerate: EI > E0 when I > 0.

• We will use real algebra in the following (non-relativistic quantum chemistry):

〈Ψ|Φ〉 = 〈Φ|Ψ〉∗ = 〈Φ|Ψ〉

• The exact ground-state energy can be expressed as E0 = minΨ〈Ψ|H|Ψ〉 = 〈Ψ0|H|Ψ0〉

where the minimization is restricted to normalized wavefunctions Ψ.

Proof: ∀Ψ, |Ψ〉 =∑I

CI |ΨI〉 and 〈Ψ|H|Ψ〉 − E0〈Ψ|Ψ〉 =∑I

C2I

(EI − E0

)≥ 0.



Variational principle for the ground state

• Note that, if Ψ 6= Ψ0, then 〈Ψ|H|Ψ〉 > E0 .

Proof:

Since Ψ 6= Ψ0, we can find a non-zero integer K such that CK 6= 0,otherwise C2

0 = 〈Ψ|Ψ〉 = 1 −→ Ψ = Ψ0 (!).

Consequently,

〈Ψ|H|Ψ〉 − E0 =∑I 6=0

C2I

(EI − E0

)= C2

K︸︷︷︸(EK − E0

)︸︷︷︸+

∑I 6=0,K

C2I︸︷︷︸(EI − E0

)︸︷︷︸ > 0

> 0 > 0 ≥ 0 > 0



Variational principle for the excited states

• Note also that the first excited-state energy E1 can be obtained variationally under normalization(〈Ψ|Ψ〉 = 1) and orthogonality (〈Ψ|Ψ0〉 = 0) constraints:

E1 = minΨ⊥Ψ0

〈Ψ|H|Ψ〉 = 〈Ψ1|H|Ψ1〉

Proof:

Since 〈Ψ|Ψ0〉 = 0 = C0, |Ψ〉 =∑I>0

CI |ΨI〉 and

〈Ψ|H|Ψ〉 − E1〈Ψ|Ψ〉 =∑I>0

C2I

(EI − E1

)≥ 0.

• Additional orthogonality constraints (〈Ψ|Ψ1〉 = 0, ...) enable to reach second and higherexcited-state energies.



Stationarity condition

• Let us consider a function f : x 7→ f(x) and the Taylor expansion around x0 through first orderin δx = x− x0:

f(x) = f(x0 + δx) = f(x0) +df(x)

dx

∣∣∣∣x=x0

× δx + . . .

• We denote δf(x0) the expansion of f(x0 + δx)− f(x0) through first order in δx:

δf(x0) =df(x)

dx

∣∣∣∣x=x0

× δx .

• x0 is a stationary point for f if δf(x0) = 0 for any value of δx.

• In this example, where f is a function, the stationarity condition readsdf(x)

dx

∣∣∣∣x=x0

= 0.

• Extrema of f (minima or maxima) are, for example, stationary points.



• Let us now consider the energy functional E : Ψ 7→ E[Ψ] = 〈Ψ|H|Ψ〉which applies tonormalized wavefunctions Ψ only.

• Note that the electronic wavefunction Ψ is a function of the electron coordinates. The energy is a"function" of Ψ,

E[Ψ] =

∫dr1 . . .

∫drN Ψ∗(r1, . . . , rN )HΨ(r1, . . . , rN ),

hence the name functional.

• The normalization condition 〈Ψ|Ψ〉 = 1 implies 〈Ψ|H − E[Ψ]|Ψ〉 = 0 .

• If we consider infinitesimal variations Ψ→ Ψ + δΨ around Ψ that preserve normalization, we haveδ〈Ψ|H − E[Ψ]|Ψ〉 = 0, thus leading to δE[Ψ] = 2〈δΨ|H − E[Ψ]|Ψ〉. Therefore

δE[Ψ] = 0 ⇔ H|Ψ〉 = E[Ψ]|Ψ〉

• Important conclusion: both ground- and excited-state wavefunctions are stationary points for theenergy functional.



Mathematical interlude: Lagrangian

• Rather than taking into account the normalization constraint 1− 〈Ψ|Ψ〉 = 0 explicitly in thederivation of the stationarity condition, it is more convenient to introduce the so-called lagrangianfunctional (or simply Lagrangian),

L[Ψ, E] = E[Ψ] + E(1− 〈Ψ|Ψ〉

),

where E , which is referred to as Lagrange multiplier, is a number that has to be determined.

• The stationarity condition can then be rewritten as

∂L[Ψ, E]

∂E= 0 → 1− 〈Ψ|Ψ〉 = 0 normalization condition !

AND

δL[Ψ, E] = 0 → 2〈δΨ|H − E|Ψ〉 = 0 for any δΨ (no constraint)

• Note that, when Ψ is stationary, E = E[Ψ].



Schrödinger equation for two-electron systems

• Two-electron wavefunction: Ψ(r1, r2)

where r1 ≡ (x1, y1, z1) et r2 ≡ (x2, y2, z2) are space coordinates of electron 1 and 2.

• Schrödinger equation: HΨ(r1, r2) = EΨ(r1, r2)

where the two-electron Hamiltonian equals H = T + V + Wee, with

T ≡ −1

2∇2

r1−

1

2∇2

r2←− kinetic energy operator

V ≡(v(r1) + v(r2)

)× ←− nuclear attraction potential operator

Wee ≡1

|r1 − r2|× ←− electron-electron repulsion operator



Schrödinger equation for two-electron systems

• For an atom with atomic number Z: v(r) = −Z

r

• For the H2 molecule: v(r) = −1

|r−RA|−

1

|r−RB |



Particular case of "non-interacting" electrons

• Let us assume that electrons do not repel each other (!) −→ Wee ≡ 0.

• If one can solve the following one-electron Schrödinger equation,

(−

1

2∇2

r + v(r))ϕ(r) = εϕ(r),

then a trivial solution to the Schrödinger equation for two electrons is

Ψ(r1, r2) = ϕ(r1)ϕ(r2) and E = 2ε .

Proof:

(T + V

)ϕ(r1)ϕ(r2) = ϕ(r2)

[(−

1

2∇2

r1+ v(r1)

)ϕ(r1)

]+ ϕ(r1)

[(−

1

2∇2

r2+ v(r2)

)ϕ(r2)

]

= ϕ(r2)εϕ(r1) + ϕ(r1)εϕ(r2)

= 2εϕ(r1)ϕ(r2)



Returning to interacting electrons ...

• Describing interacting electrons (Wee 6≡ 0) is more complicated. Indeed, in this case, any exactsolution Ψ(r1, r2) to the Schrödinger equation cannot be written as ϕ(r1)ϕ(r2):

Ψ(r1, r2) 6= ϕ(r1)ϕ(r2).

Proof : Let us assume that we can find an orbital ϕ(r) such that H(ϕ(r1)ϕ(r2)

)= Eϕ(r1)ϕ(r2)

for any r1 and r2 values. Consequently,

Wee

(ϕ(r1)ϕ(r2)

)= Eϕ(r1)ϕ(r2)−

(T + V

)ϕ(r1)ϕ(r2).

Using the definition of the operators and dividing by ϕ(r1)ϕ(r2) leads to

1

|r1 − r2|= E +

1

2

∇2r1ϕ(r1)

ϕ(r1)+

1

2

∇2r2ϕ(r2)

ϕ(r2)− v(r1)− v(r2).

In the limit r2 → r1 = r, it comes ∀ r, E +∇2

rϕ(r)

ϕ(r)− 2v(r)→ +∞ absurd !



Hartree–Fock approximation for two electrons

• A Hartree product Φ(r1, r2) = ϕ(r1)ϕ(r2) can be used as an approximation to the exactground-state wavefunction.

• The "best" ϕ(r) orbital is obtained by applying the variational principle and by restricting theminimization to Hartree products. Thus we obtain an approximate ground-state energy which isknown as the Hartree–Fock (HF) energy:

EHF = minϕ〈Φ|H|Φ〉

• Note that ϕ(r) should be normalized so that Φ(r1, r2) is normalized.

EXERCISE: (1) Show that

〈Φ|H|Φ〉 = 2

(−

1

2

∫R3

drϕ(r)∇2rϕ(r) +

∫R3

dr v(r)ϕ2(r)

)+

∫R3

∫R3

dr dr′ϕ2(r)ϕ2(r′)

|r− r′|



Hartree–Fock approximation for two electrons

(2) By introducing the Lagrangian L[ϕ] = 〈Φ|H|Φ〉+ ε(1− 〈ϕ|ϕ〉

),

show that the minimizing HF orbital ϕHF(r) fulfills the following self-consistent equation,

(−

1

2∇2

r + v(r) +

∫R3

dr′ϕ2

HF(r′)

|r− r′|

)ϕHF(r) = εHFϕHF(r).

(3) Show finally that EHF = 2 εHF −∫R3

∫R3

dr dr′ϕ2

HF(r)ϕ2HF(r′)

|r− r′|←− EHF 6= 2 εHF !

• In practice, the HF equation is solved approximately by decomposing the trial orbital ϕ(r) in a finitebasis of non-orthogonal (gaussian) atomic orbitals (AO)

{χp(r)

}p=1,...,M

:

ϕ(r) =

M∑p=1

Cp χp(r).

The so-called molecular orbital (MO) coefficients{Cp}p=1,...,M

are then optimized variationally.



0

0.1

0.2

0.3

0.4

0.5

0.6

0 1 2 3 4 5

wav

efun

ctio

n

r [a.u.]

1s orbital of the hydrogen atom

e−r/√π [E=−0.50000 a.u.]

0.28 e−0.28 r2 [E=−0.42441 a.u.]

0.2e−2.23 r2+0.19e−0.41 r2+0.06e−0.11 r2 [E=−0.49491 a.u.]

0.06e−33.9 r2+0.11e−5.1 r2+0.16e−1.16 r2+0.15e−0.33 r2+0.05e−0.1 r2 [E=−0.49981 a.u.]10 gaussians [E=−0.49999 a.u.]



0.51

0.52

0.53

0.54

0.55

0.56

0.57

0 0.02 0.04 0.06 0.08 0.1

wav

efun

ctio

n

r [a.u.]

1s orbital of the hydrogen atom close to the nucleus

e−r/√π [E=−0.50000 a.u.]

0.06e−33.9 r2+0.11e−5.1 r2+0.16e−1.16 r2+0.15e−0.33 r2+0.05e−0.1 r2 [E=−0.49981 a.u.]10 gaussians [E=−0.49999 a.u.]



Electron correlation

• Let us stress that HF is an approximate method. In the following, we shall refer to the differencebetween the exact ground-state energy E0 and the HF energy as the correlation energy Ec:

Ec = E0 − EHF < 0

• "Modelling electron correlation" means "going beyond the HF approximation".

• Note that the doubly-occupied HF orbital ϕHF(r) is an eigenfunction of the so-called Fock operator

f ≡ −1

2∇2

r +

(v(r) +

∫R3

dr′ϕ2

HF(r′)

|r− r′|

)×

• Of course, this operator has many other eigenfunctions{ϕi(r)

}i=1,2,...

with energies{εi

}i=1,2,...

that are higher than εHF. These orbitals are referred to as virtual orbitals (or just

virtuals).

• Frontier orbitals: ϕHF(r) is referred to as HOMO (Highest Occupied Molecular Orbital)and ϕ1(r) is the LUMO (Lowest Unoccupied Molecular Orbital).



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Electron correlation

• Virtual orbitals can actually be used as a basis for modelling electron correlation, thus leading to thefollowing (better) approximation to the exact ground-state wavefunction,

Ψ0(r1, r2) ≈ ϕHF(r1)ϕHF(r2) ←− HF wavefunction

+∑i≥1

Ci

(ϕHF(r1)ϕi(r2) + ϕi(r1)ϕHF(r2)

)←− single excitation

+∑

j≥i≥1

Cij

(ϕi(r1)ϕj(r2) + ϕj(r1)ϕi(r2)

)←− double excitation

• The coefficients Ci and Cij are optimized variationally.

• If the distribution of ALL the electrons (two here) in ALL the orbitals (occupied and virtuals) isconsidered, the method is referred to as Full Configuration Interaction (FCI).

• The FCI method is exact in a given finite basis of atomic orbitals. In this case, the FCI wavefunctionis of course not equal to the exact wavefunction Ψ0(r1, r2). The latter can only be reached, inprinciple, by using an infinite basis.



−1.18

−1.16

−1.14

−1.12

−1.1

−1.08

−1.06

−1.04

−1.02

−1

1.5 2 2.5 3 3.5 4

gro

und−

stat

e en

ergy

E(R

) [a

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bond distance R [a.u.]

H2 [1s=0.06e−33.9 r2+0.11e−5.1 r2+0.16e−1.16 r2+0.15e−0.33 r2+0.05e−0.1 r2]

HF (1s)

HF (1s, e−0.33 r2, e−0.1 r2)HF (aug−cc−pVQZ)

FCI (aug−cc−pVQZ)



−1.2

−1.15

−1.1

−1.05

−1

−0.95

−0.9

−0.85

−0.8

−0.75

2 3 4 5 6 7 8 9 10

grou

nd−

stat

e en

ergy

E(R

) [a

.u.]

bond distance R [a.u.]

H2 in the dissociation limit

HF (aug−cc−pVQZ)

FCI (aug−cc−pVQZ)


Documents

Introduction to advanced quantum chemistry fileIntroduction to advanced quantum chemistry Introduction to advanced quantum chemistry Emmanuel Fromager Institut de Chimie de Strasbourg